Theorem lubfun 18358

Description: The LUB is a function. (Contributed by NM, 9-Sep-2018.)

Hypothesis

Ref	Expression
lubfun.u	⊢ 𝑈 = (lub‘𝐾)

Assertion

Ref	Expression
lubfun	⊢ Fun 𝑈

Proof of Theorem lubfun

Dummy variables 𝑥 𝑠 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		funmpt 6548	. . . 4 ⊢ Fun (𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (℩𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧))))
2		funres 6552	. . . 4 ⊢ (Fun (𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (℩𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)))) → Fun ((𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (℩𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧))}))
3	1, 2	ax-mp 5	. . 3 ⊢ Fun ((𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (℩𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧))})
4		eqid 2756	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
5		eqid 2756	. . . . 5 ⊢ (le‘𝐾) = (le‘𝐾)
6		lubfun.u	. . . . 5 ⊢ 𝑈 = (lub‘𝐾)
7		biid 263	. . . . 5 ⊢ ((∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)) ↔ (∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)))
8		id 22	. . . . 5 ⊢ (𝐾 ∈ V → 𝐾 ∈ V)
9	4, 5, 6, 7, 8	lubfval 18356	. . . 4 ⊢ (𝐾 ∈ V → 𝑈 = ((𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (℩𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧))}))
10	9	funeqd 6532	. . 3 ⊢ (𝐾 ∈ V → (Fun 𝑈 ↔ Fun ((𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (℩𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧))})))
11	3, 10	mpbiri 260	. 2 ⊢ (𝐾 ∈ V → Fun 𝑈)
12		fun0 6575	. . 3 ⊢ Fun ∅
13		fvprc 6848	. . . . 5 ⊢ (¬ 𝐾 ∈ V → (lub‘𝐾) = ∅)
14	6, 13	eqtrid 2803	. . . 4 ⊢ (¬ 𝐾 ∈ V → 𝑈 = ∅)
15	14	funeqd 6532	. . 3 ⊢ (¬ 𝐾 ∈ V → (Fun 𝑈 ↔ Fun ∅))
16	12, 15	mpbiri 260	. 2 ⊢ (¬ 𝐾 ∈ V → Fun 𝑈)
17	11, 16	pm2.61i 183	1 ⊢ Fun 𝑈

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   = wceq 1554   ∈ wcel 2136  {cab 2734  ∀wral 3070  ∃!wreu 3359  Vcvv 3448  ∅c0 4280  𝒫 cpw 4549   class class class wbr 5094   ↦ cmpt 5175   ↾ cres 5642  Fun wfun 6504  ‘cfv 6510  ℩crio 7341  Basecbs 17221  lecple 17269  lubclub 18317

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-rep 5221  ax-sep 5240  ax-nul 5250  ax-pow 5316  ax-pr 5384

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ne 2952  df-ral 3071  df-rex 3081  df-rmo 3361  df-reu 3362  df-rab 3409  df-v 3450  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4475  df-pw 4551  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4945  df-br 5095  df-opab 5157  df-mpt 5176  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6512  df-fn 6513  df-f 6514  df-f1 6515  df-fo 6516  df-f1o 6517  df-fv 6518  df-riota 7342  df-lub 18352

This theorem is referenced by:  joinfval  18379  joinfval2  18380